An improved meniscus surface model for contacting rough surfaces

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Abstract

Based on the Extended-Maugis–Dugdale (EMD) elastic theory, a single asperity capillary meniscus model considering asperity deformation due to both contact and adhesive forces was developed. Specifically included in the single asperity meniscus model was the solid surface interaction inside the contact area. Subsequently, the single asperity model was coupled with a statistical roughness surface model to develop an improved meniscus surface model applicable to a wide-range of humidity levels and adhesion parameter values. Simulations were performed using typical surfaces found in microelectromechanical systems (MEMS) and magnetic storage hard disk drives to examine the effects of surface roughness and relative humidity. It was found that smoother surfaces give rise to higher adhesive and pull-off forces, and at higher relative humidity levels, the capillary force governs the adhesive behavior. As the humidity decreases, the solid surface interaction increases and needs to be included in the total meniscus adhesion. By integrating the adhesive force-displacement curves, the adhesion energy per unit area was calculated for MEMS surfaces and favorably compared with reported experimental data.

Graphical abstract

In the presence of capillary condensation, strong adhesive forces arise due to the formation of liquid menisci between proximity surfaces. An improved meniscus surface model was proposed including the effects of asperity deformations and the contribution of the solid surface interaction inside the contact zone.

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Introduction

When two surfaces approach each other, in the presence of liquid lubricant or adsorbed/trapped water, strong adhesive forces arise due to the formation of liquid menisci between proximity surfaces. In microelectromechanical systems (MEMS), capillary (or meniscus) adhesion is a common failure mechanism, a major reliability concern, and a barrier to commercializing MEMS devices [1]. Capillary adhesion could occur during the final fabrication stages involving the chemical wet-etch and liquid rinse (release adhesion), during the normal operation at high humidity environments (in-use adhesion), or while parts are stored (dormancy adhesion). In magnetic storage hard disk drives, the meniscus force between the recording slider and the rotating disk due to the existence of a molecularly thin liquid lubricant can lead to increased contact and friction forces, and result in catastrophic failures, especially for contact–start–stop type hard disk drives [2].

Significant research work has been published to theoretically and experimentally investigate the adhesion behavior between two rough surfaces in microsystems. Incorporating different single asperity models into the Greenwood and Williamson [3] (GW) statistical model, several meniscus models [4], [5], [6], [7] have been presented to predict the effects of surface roughness/texture, lubricant film thickness, and environmental humidity. However, in these multi-asperity meniscus models, the total adhesive force was determined by summing up the capillary forces over the menisci formed at the interface and other forces such as van der Waals were ignored. These simplified models are valid for high relative humidity levels (>70%) and could significantly underestimate the adhesion energy (or force) at intermediate and low humidity levels. Also, none of these models considered the change of the projected wetted area due to the asperity deformation caused by the contact and adhesive forces.

A different modeling approach to predict adhesive forces in microsystems in the presence of molecularly thin lubricants has also been proposed, using contact-mechanics based adhesion models. Incorporating the Kogut and Etsion (KE) dry contact model [8] with the original sub-boundary lubrication (SBL) [9] adhesion model, an alternative model termed improved SBL (ISBL) model [10] was presented to take into account the presence of molecularly thin lubricant. In this model, the very thin lubricant strongly and uniformly adheres to the surface and the meniscus formation is energetically unfavorable. Thus this model is not applicable for MEMS devices at high humidity levels and cannot capture the adhesion behavior in the presence of mobile lubricant in magnetic storage hard disk interfaces. Therefore, a more accurate and generalized meniscus model including contact mechanics and surface roughness effects is highly desired and proposed in this work.

To investigate the single asperity capillary adhesion, Fogden and White [11] extended the Hertz theory to analyze the contact elasticity of a single sphere contacting a rigid flat surface in the presence of capillary condensation. However, the case of noncontacting asperities was not included in their work. Based on the linear fracture mechanics, Maugis [12] derived an adhesion model for single asperity contact under dry conditions where a constant stress was assumed to act in the “Dugdale” zone outside the contact area, known as Maugis–Dugdale (MD) model. To compare the MD model with experimental data (using an atomic force microscope), Carpick et al. [13] presented a simpler general equation to approximate Maugis' solution. Employing the MD adhesion model Adams et al. [14] developed a scale-dependent friction model for rough contacting surfaces under dry conditions. However, in their model they only considered the adhesion from the contacting and intimately contacting asperities. Extending Maugis' work, Shi and Polycarpou [15] proposed a full range “dry” adhesion model (referred to as Extended-Maugis–Dugdale EMD model) considering the noncontacting asperities according to the Dugdale stress distribution assumption.

Maugis [12] and Johnson [16] pointed out that the Laplace pressure acting in a meniscus area is a “perfect” example of the Dugdale model. In this paper, the EMD theory [12], [15] was adopted to analyze the single asperity adhesion behavior in the presence of capillary condensation. Instead of the approximation to the Lennard-Jones potential, in this paper, the Dugdale stress is taken as the Laplace pressure acting on the wetted area. For contacting asperities, the asperity deformation due to contact and attractive forces as well as the adhesive force inside the contact region was considered. For noncontacting asperities, unlike existing meniscus models such as the classical equation presented by Israelachvili [17], the spherical deformation due to the capillary force was also taken into account. A proposed roughness parameter, termed limiting asperity height hasp, was presented to separate the asperities into five types based on their height above the mean plane for a given relative humidity. Also, equations to predict the adhesive forces for each asperity type were proposed. The EMD-based single asperity model was subsequently incorporated into a statistical roughness model to develop an improved meniscus rough surface model for a wide-range of relative humidity levels. Simulations were performed for realistic MEMS surfaces and magnetic storage head disk interfaces. By integrating the force-displacement curve, the adhesion energy between the polysilicon MEMS structures and the substrate was determined and compared with reported experimental data.

Section snippets

Improved single asperity meniscus model

The static contact between two rough surfaces can be modeled as the contact between an equivalent rough surface and a flat rigid surface. In the presence of capillary condensation, menisci form around the asperities due to surface tension effects, as shown schematically in Fig. 1. By integrating the contribution of each asperity, the adhesion force and the contact load between the two rough surfaces can be determined. To calculate the single asperity capillary force, the surface asperities can

Meniscus surface roughness model

To predict the adhesive force between two rough surfaces at different relative humidity levels, the EMD-based single asperity meniscus model was incorporated into the GW-based statistical surface roughness model, which assumes that two rough surfaces can be represented by an equivalent rough surface in contact with a smooth surface [3], as shown schematically in Fig. 4. For such a multi-asperity contact interface, the net adhesive force arises from the individual asperity contributions. To

Pull-off force

Simulations were performed with the parameter values listed in Table 2, which represent typical surface micromachined polysilicon structures on a silicon nitride substrate with σ=1.927nm, R=0.87μm, and η=26.83μm−2. The fabrication process of these samples and the details of the AFM measurements were described in Ref. [22].

Fig. 5 shows rough surface adhesive forces as a function of surface separation for relative humidity values of 0.4, 0.6, and 0.85 (corresponding to λ values of 0.22, 0.12,

Comparisons with experiments

In MEMS, meniscus adhesion is a common failure mechanism and a major reliability concern. Several experimental techniques have been proposed to study the adhesion between two MEMS surfaces. Most experiments used microcantilever beams to determine the adhesion energy, instead of the adhesion force, between the MEMS structures and the substrate based on a fracture mechanics approach [27], [28], [29]. To compare the proposed model with published MEMS adhesion experiments, the force-displacement

Summary

An improved meniscus surface model valid for a wide range of relative humidity values and based on the elastic EMD theory and specifically including asperity deformation due to capillary adhesion solid surface adhesion and surface roughness was presented and favorably compared with published experimental data. The main findings are:

  • (1)

    Using the elastic EMD theory, the adhesion behavior of a single asperity on a rigid flat in the presence of capillary condensation was analyzed. It shows that due to

Acknowledgements

This research was supported by the National Science Foundation under grants number CAREER CMS-0239232. The authors gratefully acknowledge this support.

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